Transcript Probability: Many Random Variables (Part 2)

Probability: Many Random
Variables (Part 2)
Mike Wasikowski
June 12, 2008
Contents
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Indicator RV’s
Derived RV’s
Order RV’s
Continuous RV Transformations
Indicator RV’s
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IA = 1 if event A occurs, 0 if not
Consider A1, A2, …, An events, I1, I2, …, In their
indicator RV’s, and p1, p2, …, pn the
probabilities of events Ai occurring
Then Σj Ij is the total number of events that
occur
Mean of a sum of RV’s = sum of the mean of
the RV’s (regardless of dependence), so E(Σj
Ij) = Σj E(Ij) = Σj pj
Indicator RV’s
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If all values of pi are equal, then E(Σj Ij) =
np
When all events are independent, we
calculate variance of number of events
that occur as p1(1-p1)+…+pn(1-pn)
If all values of pi are equal and all events
are independent, variance is np(1-p)
Thus, we have a binomial distribution
Ex: Sequencing EST Libraries
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Transcription: DNA → mRNA → amino acids/proteions
EST (expressed sequence tag): a sequence of 100+ base
pairs of mRNA
Different genes get expressed at different levels inside a
cell
Abundance class L: where a cell contains L copies of an
mRNA “species”
Generate an EST DB by sampling with replacement from
the mRNA pool, see less rare species less often
How does the number of samples affect the proportion of
rare species we will see?
Ex: Sequencing EST Libraries
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Using indicator RV’s makes this problem
easy to solve
Let Ia = 1 if a is in the S samples, 0 if not
Number of species in abundance class L
= Σa I a
We know each Ia has the same mean, so
E(Σa Ia) = nLpL
Ex: Sequencing EST Libraries
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Let pL = 1-rL, where rL is the probability
this species is not in the database
rL = (1-L/N)S
Thus, we get E(Σa Ia) = nL(1- (1-L/N)S)
Derived RV’s
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Previously saw how we find joint distributions
and density functions
These joint pdf’s can be used to define many
new RV’s
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Sum
Average
Orderings
Because many statistical operations use these
RV’s, knowing properties of their distributions
is important
Sums and Averages
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Two most important derived RV’s
Sn = X1+X2+…+Xn
X = Sn/n
Mean of Sn = nμ, variance = nσ2
Mean of X = μ, variance = σ2/n
These properties generalize to well-behaved
functions of RV’s and vectors of RV’s as well
Many important applications in probability and
statistics use sums and averages
Central Limit Theorem
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If X1, X2, ..., Xn are iid with a finite mean
and variance, as n→∞, the standardized
RV (X-μ)sqrt(n)/σ converges to an RV ~
N(0,1)
Image from Wikipedia:
Central Limit Theorem
Order Statistics
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Involve the ordering of n iid RV’s
Call smallest X(1), next smallest X(2), up
to biggest X(n)
Xmin = X(1), Xmax = X(n)
We know that these order statistics are
distinct because P(X(i) = X(j)) = 0 for
independent continuous RV’s
Minimum RV (Xmin)
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Let X1, X2, ..., Xn be iid as X
If Xmin ≥ x, then for each Xi, Xi ≥ x
P(Xmin ≥ x) = P(X ≥ x)n, also written as 1Fmin(x) = (1-FX(x))
By differentiating, we get the density
function
fmin(x) = n fX(x) (1-FX(x))n-1
Maximum RV (Xmax)
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Let X1, X2, ..., Xn be iid as X
If Xmax ≤ x, then for each Xi, Xi ≤ x
P(Xmax ≤ x) = P(X ≤ x)n, also written as
Fmax(x) = (FX(x))n
By differentiating, we get the density
function
fmin(x) = n fX(x) (FX(x))n-1
Density function of X(i)
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Let h be a small value, ignore events of
probability o(h)
Consider the event that u < X(i) < u+h
In this event, i-1 RV's are less than u, one is
between u and u+h, the remaining exceed u+h
Multinomial event with n trials and 3 outcomes
We have an approximation of P(u < X < u+h) ~
fX(u)h
Density function of X(i)
Continuous RV Transformations
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Consider n continuous RV's, X1, X2, ..., Xn
let V1 = V1(X1, X2, ..., Xn), V2, ..., Vn defined similarly
we then have a mapping from (X1, X2, ..., Xn) to (V1, V2, ..., Vn)
If the mapping is 1-1 and differentiable with a differentiable
inverse, we can define the Jacobian matrix
Jacobian transformations are used to find marginal functions of
one RV when that would be otherwise difficult
Used in ANOVA as well as BLAST
Questions?